Final answer:
To estimate the instantaneous rate of growth in 2007 and compare it with 2006, we use derivative analysis to calculate the derivative of the population for each year. The instantaneous growth rate is given by the equation dN = (b - d)N, which simplifies to rN. For extended periods of economic growth rates like GDP, averages are used as denominators for calculations.
Step-by-step explanation:
To estimate the instantaneous rate of growth in the year 2007 and compare it with the growth rate in 2006, we need to apply derivative analysis, which is a part of calculus dealing with the rate at which quantities change. The instantaneous rate of growth is the derivative of the population with respect to time (dN/dt), which represents the change in population at a specific moment in time. The derivative of the population, dN = (b - d)N, where 'b' is the birth rate per capita, 'd' is the death rate per capita, and 'N' is the number of individuals, simplifies to 'rN', where 'r' is the intrinsic rate of increase.
When comparing the instantaneous growth rate between two different years, we calculate the derivative for each year separately and then analyze the change. As for comparing growth rates with GDP data, the growth rate can be computed differently for a one-time period versus over extended periods where it might be better to use the average quantity over the period in question as a denominator. Nonetheless, the fundamental concept of using derivatives to determine instantaneous rates remains constant whether we're discussing populations or economic figures such as GDP.